Hilbert Function
Let 
be a collection of 
distinct points. Then the numberof conditions imposed by 
on forms of degree 
is called the Hilbert function 
of .If curves 
and 
of degrees and 
meet in a collection of 
points, then for any 
,the number 
of conditions imposed by on forms of degree is independent of and andis given by
where the Binomial Coefficient
is taken as 0 if 
(Cayley 1843).
References
Eisenbud, D.; Green, M.; and Harris, J. ``Cayley-Bacharach Theorems and Conjectures.'' Bull. Amer. Math. Soc. 33, 295-324, 1996.
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